Recently, quantum phase transition [1]} has received
great attention due to its intimate correlation to the fundamental
principles of quantum mechanics, especially to the concept of
quantum entanglement [2]}, [3]}, [4]}, [5]} ( see
Ref.[6]} for a comprehensive review ). In general the
quantum phase transition happens when the degeneracy of the ground
states occurs, which cannot be characterized completely by the
pattern of symmetry broken ( order parameters of some kind ).
Instead the universal quantum order or topological order is needed
for the description of properties of the ground state in many-body
systems [7]}. Recently the connection between the geometric
phase of the ground state and the quantum criticality has been
displayed in spin-chain systems by displaying the singularity of the
geometric phase closed to the critical points [8]}, [9]}.
Moreover in Ref [9]} the author showed that the scaling
behavior of the geometric phase of the ground state near the
critical points can also display the universal class of the phase
transitions. Many works have been devoted to this interesting issue
[11]}, [12]}, [13]}, [14]} ( or see Ref.[15]} for a
review ).
